Limit distributions and scaling functions

TL;DR

This paper reviews the asymptotic behavior of lattice polygon models, focusing on limit distributions and scaling functions, especially in the uniform perimeter ensemble, providing geometric insights into their scaling behavior.

Contribution

It offers a pedagogic review connecting limit distributions to the scaling behavior of generating functions for lattice polygons.

Findings

Limit distributions relate to the scaling of perimeter and area generating functions.

Provides geometric interpretation of scaling functions.

Summarizes known results on asymptotic behavior of lattice polygons.

Abstract

We discuss the asymptotic behaviour of models of lattice polygons, mainly on the square lattice. In particular, we focus on limiting area laws in the uniform perimeter ensemble where, for fixed perimeter, each polygon of a given area occurs with the same probability. We relate limit distributions to the scaling behaviour of the associated perimeter and area generating functions, thereby providing a geometric interpretation of scaling functions. To a major extent, this article is a pedagogic review of known results.